Anyone who thinks you owe him money or injured him can sue you in court. If the judge finds you personally liable, the injured party may be able to put a lien on your house, garnish your wages or suck out the contents of your bank account to settle the debt. Personal liability insurance protects you by paying off the court judgment without tapping your assets.
If you're a homeowner, you probably have personal liability insurance already. It's part of the standard homeowners policy, along with the property insurance that pays for damage to your house. And no, it's absolutely not tax deductible. The IRS says specifically that insurance on your personal home isn't a write-off -- the only insurance costs you can deduct are any premiums you pay for mortgage insurance. Insurance for your home is a personal expense, and the IRS doesn't let you deduct many of those.
If you own a rental property, insurance on the rental is fully deductible as a business expense. Taking out liability insurance along with property insurance is a good idea, as it protects your assets if a tenant slips on a rickety step or says you evicted her illegally. You deduct rental expenses from rental income. If you have red ink left over and you actively manage the rental, you can deduct up to $25,000 from other income.
If you have a small business or work as an independent contractor, you might want to take out personal liability business insurance. This protects your personal assets if, say, a client sues you for screwing up. The personal liability in a regular homeowners policy doesn't protect you if you're sued over a business activity. You can pay extra to expand your homeowners coverage or take out a separate business-liability policy. Either way, it's a business expense you can deduct from your business income.
If you add business coverage to your homeowners policy, or take out an umbrella liability policy -- one that covers you when your other insurance maxes out -- you can't claim the entire policy as a write-off. You have to prorate the part of your premium that pays for business or rental-related liability coverage and report that as an expense. Likewise, if you rent out your vacation home, you have to prorate the insurance for periods you're staying there. If, say, you and your family use it for one month a year, you only get to write off 11 months' worth of premiums.